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16—47372-1 QPO 



W A L T H A M 



WATCH PAPERS 



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WALTHAM 



WATCH PAPERS. 



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I. 






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ALFRED MUDGE & SON, PRINTERS, 
34 School Street, Bostox. 



PREFATORY. 



The American Watch Company, of Waltham, proposes to issue for 
gratuitous distribution to its customers and the watcli trade, a series of 
small pamphlets, which shall contain such essays, new and old, on the 
theory and practice of watch making, and such selections of interesting 
and valuable matter from horological works and journals as may seem 
best suited to the end the Company has in view ; namely, to excite a 
deeper interest in sound and scientific watch making in this country, and 
incidentally, by the better cu-culation here of the best foreign treatises on 
the nicer points of the art, to give and get information. The Company, as 
is well known, has disregarded entirely foreign precedent in its method of 
manufacturing, and has even released itself, by investigation and experi- 
ment, from many practices which seem to involve principles, and, as the 
managers think, with the best results ; still, there is a vast fund of stand- 
ard knowledge in foreign works, upon the basis of which all good watch 
making, here and there, must proceed. Additions to this stock are, 
besides, being constantly made and published in foreign countries for the 
benefit of all laborers m the art. It is to give this knowledge a freer 
diffusion in our own coiintry, where comparatively little has been pub- 
lished tipon these topics, and where each man is too often his own teacher, 
and, by provoking inquiry and criticism, to make some advance in the 
delicate business it pursues, that the Company imdertakes this little pub- 
lication. 

The pamphlet may be the medium, too, from time to time, of any 
communications the Company may wish to make to its customers or the 
public, and will be open to correspondence and advertisements from any 
source related to the trade. 

Waltham, Mass., December. 1862. 



ISOOHRONISM OF BALANOE-SPEINGS. 



On the Laws of Isochronism of the Balance Spring, as connected 
with the higher order of Adjustments of Watches and Chro- 
nometers. By Charles Frodsham, Assoc. Inst. C. E. 
•^ \\ 

There is no subject connected with the science of horology, 
upon which such general deficiency of knowledge prevails, as that 
relating to the laM's of the isochronism of the balance-spring, in 
connection Avith the higher order of adjustments of watches and 
chronometers ; and, however surprising the circumstances may 
appear, that such is the fact, is abundantly proved by the difficulty 
almost universally experienced, in procuring competent assistants 
in the higher branches of the art. To explain these principles, 
and difi^use this knowledge, therefore, is the object of the present 
Paper, and in laying it before the Members of the Institution of 
Civil Engineers, the Author hopes to do justice to the talents of 
the watch-makers of the eighteenth century, whose numerous 
researches and inventions constitute the basis of all the horologi- 
cal knowledge possessed at the present day; for both in the best 
chi-onometers and watches, the principles of the makers of that 
period are still invariably followed. It is true, that the separate 
pieces in both, are now, by the aid of machinery and by the prac- 
tical skill of the workmen, produced in a very high state of 
perfection ; but it is not less true, that although the division of 
labor has contributed to insure the perfection of the parts, sep- 
arately considered, horology, as a science, has not advanced 
proportionably ; and talanted individuals are become more scarce, 
who, by the study of its laws, are qualified for the task of seeing 
that all its principles are properly carried out, and the several 
well-made parts combined together, into a correct machine for 
measurino- time. 



b ISOCHRO>"IS:Sl OF BALANCE-SPRINGS. 

To regularly educated chronometer-makers, the Author does not 
pretend to offer anything new, beyond the scientific investigation 
and explanation of the principles of the isochi'onal adjustment, 
which, in a practical point of view, must be familiar to them. 
Watch-makers, however, properly so called, constitute a much 
more numerous class than that of chronometer-makers ; and, 
although the practical knowledge of the isochronal adjustment, is 
not less necessary to the former, than to the latter, at least if they 
aspire to the manufacture of watches of a superior class, yet few 
of them have attained a full knowledge of the subject, either in 
principle or practice. To another, and a far larger class, namely, 
the wearers of watches, the Author ventures to address himself, 
with a view to impart to them the requisite knowledge for dis- 
criminating between good and indifferent watches. 

Of all the adjustments necessary in the parts of a good watch, 
the most essential to its performance, is unquestionably that of 
the isochronism of the balance spring ; for, if this adjustment be 
wanting, wliatever may be the excellence of the machine in other 
respects, and however labored its workmanship and other adjust- 
ments, it will assuredly disappoint the expectation of the artist, 
who will find it incapable of being regulated to preserve the same ' 
rate of going, in the various positions in which it is liable to be 
placed. An example is subjoined, illustrative of the effects of a 
non-isochronal spring upon an otherv^ise good watch. 

Suppose, for instance, that by comparison with a good clock, 
the going of a well-made watch is tried during twelve hours, in 
four vertical positions, wherein the friction is much increased, and 
the arc of vibration of the balance considerably diminished in 
extent (those positions being with the hours xii, vi, ix, and in, 
consecutively upwards, during three hours each), and that it keeps 
correct time in all those positions ; but that in the horizontal posi- 
tion, or with the face upwards, with longer arcs of vibration, the 
watch gains one hundred and twenty seconds in twelve hours, the 
friction is least in the horizontal position, and the arcs of vibra- 
tion are consequently of the greatest extent. 

Here then is a watch, which, though gaining considerably in the 
long arcs of vibration, indicates, nevertheless, a very near approxi- 
mation to perfection, and by its correct performance in the verti- 



ISOCHRONIS.M OF BALANC'E-SPRINC^S. 7 

cal positions, shows that the balance has been most correctly 
poised. The proper remedy, in such a case, is to make a correct 
isochronal adjustment of the balance spring. A person, however, 
who is unacquainted with this adjustment, would fail to discover 
what the true remedy should be, and would follow the plan usually 
resorted to, in which by lightening the balance, at the twelve 
o'clock part, the times of the vibrations, in the hanging and lying 
positions of the watch, may be accommodated to each other ; but 
not without increasing the errors in the other three vertical posi- 
tions, to tlie great detriment of a nearly perfect watch : thus it is 
that many watches, which are fair specimens of workmanship, are 
frequently deteriorated by false adjustments, and fail to procure 
for their makers, either credit or satisfaction. 

Down to the middle of the seventeenth century, horology could 
only be considered as a mechanical art, depending entirely upon 
good workmanship for its excellence ; but at that period, Dr. 
Hooke raised it to the rank of a science, by propounding its laws, 
and enriching it with those valuable discoveries and inventions, 
which rendered skilful manipulation a mere accessory, although 
an indispensable one, to the carrying out the governing laws and 
principles, which he had deduced from the highest branches of 
science. 

The extraordinary talents of Dr Hooke as a mechanician, can- 
not be too highly admired ; for the improvements which, through 
him, were effected both in watches and clocks, do not seem to be 
so much the result of a happily conducted train of experiments, 
to which chance had directed him, as to have been elicited by 
acute reasoning, upon facts deduced from careful observation. 

Dr. Hooke first suggested the plan of reducing the vibrations 
of the pendulum of a clock, within those small circular arcs, 
wliich do not differ sensibly from the cycloidal curve, and also 
giving such an amount of momentum to the pendulum, as should 
nullify the effect of any differences that might exist, in the trans- 
mission of the impulse through the medium of the train. Great, 
however, as were these improvements in the pendulum clock, they 
bear no comparison, in value and importance, with his invention 
and application of the balance spring to the watch, which unques- 
tionably laid the foundation of the chronometric art: for hence- 



O ISOCHRO]NISM OF BALANCE-SPRINGS. 

forward this balance, with its spring, was destined to perform 
an office in the watch, equivalent to that of the pendulum of a 
clock. 

Previously to this, the watch was a machine too subject to 
iiTegularities to be relied on ; but it now became the model of a 
movable time-keeper, that by successive gradations of improve- 
ment would attain a high degree of perfection, and ultimately 
contain M'ithin itself, as at the present day, the capability of cor- 
recting its own imperfections, by means of certain applications 
of compensation and adjustment. 

It is evident, that Dr. Hooke's inventive genius, which suggested 
to him the spiral spring, penetrated through the obscm-ity which 
concealed the laws of its isochronism from those who afterwards 
employed it. His expression, "As is the tension, so is the force," 
clearly demonstrates, not only that he was acquainted with the 
isochronal property of the spring, but that the correct interpreta- 
tion of the phrase should have unfolded the law to others; it is, 
therefore, remarkable, that the spiral spring should have been 
employed for nearly a century, before any of his numerous follow- 
ers rediscovered the means of isochronizing it. 

It may be reasonably inferred, from a variety of circumstances, 
that Harrison was unacquainted with the isochronal property of 
the spiral; Arnold, however, his immediate follower, seems to 
have practically comprehended the subject, and while occupied in 
researches as to the means of lessening the difficulties of the 
operation, he invented the cylindrical form of balance spring and 
compensation balance, which were discoveries of such gi-eat 
importance in the progress of ckronometric improvement, that they 
may be said to have formed the commencement of a new era in 
the science. 

The merit of the discovery of isoclrronism in France, was con.- 
tested by the rival artists Le Roy and F. Berthoud, by the latter 
of whom the subject was, among much other very valuable infor- 
mation, elaborately treated in his " Traite des Horloges Marines," 
published at Paris in 1773. It, however, unfortunately happens, 
that the artists of the present day are too little acquainted with 
the writings and performances of those of the eighteenth centui-y, 
such as Sully, Graham, Harrison, Camus, Le Roy, Berthoud, 



ISOCBTRONISM OF BALANCE-SPRINGS. » 

Ellicott, Cumining, Mudge, Arnold, Earnshaw, etc., etc. If a 
society of persons professing the art had been formed, and papers 
on the subjects connected with its improvement had been occa- 
sionally read at the periodical meetings of its members, it is 
impossible to say to what degree of perfection the art might now 
have attained, what sums would have been spared, that have been 
squandered in useless patents, and what valuable time saved, 
which has been thrown away in making researches, the results of 
which had been long known and amply described in the works 
published about that period. 

One of the experiments performed by Berthoud must be men- 
tioned, as it has been of the greatest service in enabling the 
variations of rate to be traced to their true causes, namely, the 
changes of the elastic force of the balance spring, under changes 
of temperatm-e. 

The diminution of elastic force in balance springs by heat, was 
suspected as early as 1747; as appears from the following passage 
in the prize essay of the celebrated geometrician, Daniel Bernoulli, 
read before the French Academy : 

" I must not omit a circumstance which may be prejudicial to 
balance watches ; it is, that experimental philosophers pretend to 
have remarked that certain changes of elastic force uniformly 
follow changes of temperature. If that be the case, the spring can 
never uniformly govern the balance." 

That which Bernoulli only conjectured in 1747, was, in 1773, 
established as a matter of certainty, and the amounts in loss of 
time, due to each of these three causes, operating conjointly were 
subjected to calculation and experiment by Berthoud; with the 
following result : 

One of his marine watches in passing from 0° to 27" Reaumur 
(32° to 92° Fah.)— 

Seconds. 
Loss per diem by expansion of diameter of balance ... 62 
Ditto ditto by loss of spring's elastic force .... 312 

Ditto ditto by elongation of the spring 19 

393 or 6m. 33b. 



10 ISOCHRONISM OF BALANCE-SPRINGS. 

Few of the watch-makers of the present day can form an ade- 
qxiate notion of the difficulties which had to be overcome by the 
early watch-makers, on finding that their watches varied to so 
great an extent as six minutes thirty-three seconds in passing 
from 32° to 92° Fahrenheit : while a clock, with a seconds' pen- 
dulum, was known not to alter more than about twenty seconds, 
under similar circumstances. 

Isochronism is an inherent property of the balance-spring, 
depending entirely upon the ratio of the spring's tension, follow- 
ing the proportion of the arcs of inflection ; a balance spring, 
therefore, of any force whatever, having the progression required 
by the law of isochronism, will preserve this property, whether it 
be applied to a balance making quick, or slow vibrations ; for 
which reason, in the present inquu-y, every cu'cumstance is pur- 
posely omitted, Avhich gives to the balance its specific character, 
as Aveight, diameter, etc , and it is treated simply as the balance. 

Writers on isochronism appear to have considered the vibra- 
tions of the balance in its totality, and to have reasoned for the 
most part on the times of vibrations in their entirety ; but a 
better plan, it is submitted, Avould be, to consider the time of 
each semi-vibration of the balance, to consist of some number of 
very minute equal portions of time, and then by applying the 
known laws of forces to the motion of the balance, to determine 
what are the specific conditions under which the vibrations them- 
selves shall, in their totality, be isochronous. 

The elastic force of balance springs belongs to that class of 
forces called continuous, because the action is not by a single 
impulse, which then ceases ; but by a number of consecutive 
impulses, following each other in such rapid succession, as to con- 
stitute an vminterrupted and continuous force ; but which force is 
uniformly increasing during the bending of the spring, and 
uniformly decreasing whilst it is unbending. 

The first step towards the comprehension of isochronism, is the 
recognition of the accelerated and retarded motion of the balance ; 
for which purpose it must be followed, step by step, through the 
entire vibration, upon the supposition, that the time of each semi- 
vibration is divided into, or composed of, any convenient number 
of equal parts, as, for instance, ten. If then the balance be sup- 



ISOCHRONISM OF BALANCE-SPRINGS. 11 

posed to be moved by the finger from the position where it will 
stand, when at rest, over an arc of any number of degrees, and 
be there held, it will be perceived, that the spring is wound into 
tension, and has acquired an amount of elastic force, proportion- 
ate to the angle over which it is inflected, which force is then 
reacting against the finger, by which the balance itself is lield in 
a state of rest. 

The instant, however, that the finger is withdrawn, the elastic 
force of the spring will be exerted in overcoming the absolute 
inertia of the balance, and at the expiration of the first short 
period of time (or one-tenth of the time of a semi- vibration), the 
spring will have communicated a slight motion to the balance. 
During the second tenth, the spring's force is exerted against the 
balance in motion, instead of being at rest, as it was at the com- 
mencement of the fixst tenth, and the spring will necessarily 
accelerate the motion that the balance had previously acquired ; 
and so on during the third, fourth, and every other succeeding 
tenth, the elastic force of the spring, though continually decreas- 
ing, will be urging the balance forward, and will therefore continue 
to accelerate it, until the spring arrives at the position of quies- 
cence, where it ceases further to urge the balance. 

The balance having thus returned to the position of rest, from 
whence it was moved by the finger, the first half of the vibration 
is fully completed, and a change of circumstances takes place ; 
the spring, which continued to communicate motion to the balance 
until now that the whole of its force has been transferred thereto, 
has resumed for an instant a state of quiescence. The balance 
has also assumed a new character, having acquired a sufficient 
velocity of motion and momentum, to carry it through the other 
half of the vibration, and in so doing to inflect the spring through 
an angle, equal to that which was originally moved through by 
the finger, and to give the spring the requisite tension for perform- 
ing the next succeeding vibration. During the first few tenths of 
the second half of the vibration, the spring has so little tension, 
that its force retards but slightly the motion of the balance ; but 
during the succeeding tenths, the tension gradually increases, until 
at length the spring acquii-es sufficient force to entirely arrest the 
motion of the balance, at the same extent of arc on the other side 



12 



ISOCHRONISM OF BALANCE-SPRINGS. 



of the place of quiescence, as that to which it was originally 
moved by the finger. 

Diagram, Fig. 1, shows the magnitude of the several arcs, 
traversed by an index, affixed to the rim of the balance, during 
each of the successive tenth portions of the time, into which a 
semi-vibration of 175 degrees is conceived to be divided. 

Fig. 1. 
Point of Rest. 




An inspection of the figure, and a comparison of the spaces 
described during the first and last tenth of each semi-vibration, 
will suffice to show that each vibration of the balance is composed 
of an alternately accelerated and retarded motion, and how rapidly 
the ratio between them proceeds, although it is not probable that 
the eye could detect that to be the case, even in a balance of the 
slowest motion. 

The specific conditions under which the vibrations themselves, 
considered in their totality, whether long or short, should be 
isochronous, are these : 



ISOCHRONISM OF BALANCE-SPRINGS. 13 

1st. If the time of each semi-vibration be conceived to be com- 
posed of the same number of very small equal instants of time, 
and, whatever be the extent of the arc traversed, that the first and 
last of these minute instants of time precisely correspond with 
the commencement and conclusion of each semi-vibration, the 
vibrations, whether long, or short, of such a balance, will be 
isochronous, or be performed in equal times. 

2nd The elastic force of a balance-spring, increases in direct 
proportion to the angle of inflection, by which it is wound into 
tension ; and hence it is obvious, that the increasing and dimin- 
ishing tension, which causes the balance to follow a definite law 
of acceleration and retardation, must itself also follow some 
definite ratio of increase and decrease, in order that the first and 
last of the very small equal instants of time, shall correspond 
with the commencement and conclusion of each semi-vibration. 

3d. It is likewise evident, that the ratio of change in the 
tension, may be either one which proceeds too rapidly, and, con- 
sequently, produces an operation in excess, or one which proceeds 
too slowly, and produces an operation in defect; on which account, 
there are two varieties of spring which do not produce isochro- 
nous vibrations. 

4th. In the former variety, producing an operation in excess, 
the spring acquhes a greater amount of elastic force, than that 
which is due to the angle of inflection in an isochronal spring ; 
whence it follows, that the greater the arc of vibration, the gi-eater 
will be the angle of inflection, and, consequently, the gi-eater the 
excess of undue tension. The eff"ect of this undue tension will 
be, to hurry the balance forward during the first half of the 
vibration, with too great celerity, and thus cause it to arrive at 
the conclusion, before the complete number of minute instants, 
due to the isochronous vibration, has expu-ed. A similar eff'ect is' 
produced during the second half of the vibration, by the undue 
excess of tension arresting the balance, before the full number of 
instants has entirely expired. During each semi-vibration, there- 
fore, thi'oughout the day, some of these minute instants will be 
left unemployed, and their accumulation will be the amount 
gained in the long arcs of vibration, in comparison with the per- 
formance of the same watch in the short arcs. 



14 ISOCHRONISM OF BALANCE-SPRINGS. 

5th. In the latter variety, the elastic force due to the angle of 
inflection will not be sufficiently great, and the spring will not 
have the requisite tension to carry the balance over the first semi- 
vibration of a long arc, in the number of instants allotted to it, 
nor to arrest it so soon as the isochronous term of the second 
semi-vibration requnes. Each semi-vibration, therefore, will 
occupy too large a number of instants in its performance, and the 
accumulated amount of them, throughout the day, will indicate 
the loss during the long arcs of vibration, in comparison with the 
performance of the same watch in the short arcs. 

It is evident, that however gi-eat may be the science displayed 
in the application of a balance-spring, it will be valueless in a 
chronometrical point of view, if it will not remain permanently 
in the state in which the artist leaves it. For a spring to possess 
this indispensable property, a high degi'ee of perfection is, neces- 
sarily, requhed, demanding care in the selection of the material, 
skill in the manufacture, and science in the application. 

Balance-springs are, for the most jsart, made of steel, hardened 
and tempered, though some few have been made of gold, of 
which metal certain alloys have been particularly recommended ; 
but theii- elasticity is not always to be relied on. The use of 
glass for springs was suggested by Berthoud, but was, ultimately, 
rejected. 

Balance-springs must possess as perfect and as permanent a 
degree of elasticity as can be attained. These requisites depend 
upon the quality, hardness, and temper of the metal, as well as 
upon the form or shape of the spii-al. A soft spring gradually 
changes its form, and, losing a portion of its elastic force, becomes 
defective, and unfit for employment in a watch of the slightest 
pretensions, owing to the constant losing on its rate. A hard- 
ened and tempered spring, on the contrary, has a tendency to 
gain on its rate; but this must not be considered as a defect, 
since it is merely the result of the laminae of the spring having 
been set in the process of hardening, whereby it has acquired an 
adventitious degree of rigidity. This rigidity, however, wears off 
after a few months' vibration in the watch, which during this 
period almost imperceptibly accelerates a little upon its rate, on 
account of the acquisition of the minute increments of additional 



ISOCHRONISM OF BALANCE-SPRINGS. 15 

elastic force occasioned by the gradually increased flexibility of 
the spring. When, however, the processes of hardening and tem- 
pering have been properly conducted, the gaining on the rate will 
be restricted within very narrow limits, and will soon cease, on 
the springs' attaining its maximum amount of flexibility and 
elastic force. 

Correctness of form, or shape, has been already stated as one of 
the conditions requisite to insure isochi'onism. There are two 
form.s of springs in use, namely, the cylindrical, or helical spring, 
and the spii-al, or flat spring ; the former is exclusively employed 
in chronometers, and the latter in all other kinds of watches.*'' 

The simplest form of spring is the helical, cylindrical. This 
spring is formed into a coil of a certain number of rings of equal 
diameter, rising one above another, in the form of a cylinder. 
The lower end of the spring is turned in by a suitable curve to 
accommodate it to the size of the collet, into which that end is 
fixed ; and the upper end of the spring is turned in by a more or 
less bold sweep, according to the indications of the isochronal 
adjustment, and is pinned into a fixed stud. 

The collet vibrating with the balance, that point in the circxim- 
ference of the collet Avhere the spring is pinned into it, is 
inflected through the same extent of arc as the semi-vibration 
consists of; and by examination of the action of the spring, 
during the vibration of the balance in the chronometer, it will 
be perceived, that for each portion of the extent so inflected, 
there is a corresponding increase, or diminution of each of the 
coils of the helix, throughout the entire length of the spring, 
no part whatever being out of action during any portion of the 
vibration. 

* Writers mention a third, or spherical form, which is stated to be better adapted 
for isochronous vibrations ; but it needs the testimony of more numerous experiments 
and stronger evidence than has been hitherto adduced, to prove that chronometers with 
spherical springs are superior to those with cylindrical springs, of which so many fine 
specimens, by various makers, are now in constant use. It may likewise be observed, 
as a proof, that the flat spring is capable of the same degree of perfection as the cylin- 
drical, that the Author's prize chronometer, No. 1, which has never been surpassed by 
any chronometer on record, had a flat spring. This chronometer, during a trial of 
twelve months' duration at the Royal Observatory, at Greenwich, was found by actual 
daily observation to have made an extreme variation of only fifty-seven hundredths of 
a second. 



16 ISOCHRONISM OF BALANCE-SPRINGS. 

In order to try the isochronism of a spring, the chronometer 
must be ready and in going order. If the force of the main 
spring be then increased by setting up the ratchet, the arc of 
vibration of the balance will be extended ; or if the force of the 
main spring be diminished by letting down its ratchet, the arc 
will be lessened ; the arc of vibration may therefore be regulated 
to any extent desired. Comparisons of the chronometer's rate of 
going, dm'ing an equal number of hours, in the long and in the 
short arcs of vibration, are then to be made by a good clock, and 
the difference, if any, carefully noted. This difference indicates 
the state of approximation of the spring to isochronism, and 
points out the remedy, if it needs correction, according to the 
following rules : 

Case 1st. If the clu'onometer be found to lose in the long 
arcs, it will prove that the tension, or elastic force of the spring, 
has not increased to the amount due to the angle of inflection, or 
semi-arc of vibration. Hence, some minute portions of time are 
lost in each semi-vibration ; in the first semi-vibration, the bal- 
ance not being carried forward with sufficient celerity, so as to 
arrive soon enough at its term ; and in the second semi-vibration, 
by the spring not acquiring the requisite amount of force soon 
enough to stop the balance, at the expiration of the number of 
instants due to the isochronous semi -vibration. 

The remedy in this case is to shorten the spring, and thereby 
to cause the progression of the increase of its elastic force to 
become more rapid ; but, as much time is lost by repeatedly un- 
pinning the spring, the effect of shortening may be produced 
artificially, when the state of the isochronism is within the limits 
Avhich experience points out, by merely altering the form of the 
upper curve, so as to give it a greater degree of expansion. 

Case 2d If the chronometer is found to gain in the long arcs, 
in comparison to the time it keeps, when vibrating in short arcs, 
it will prove, that the tension increases in a ratio beyond that 
which is due to the angle of inflection. In this case, if the 
chronometer keeps time, when the semi-arc of vibration is one 
hundred degrees, it will gain when it is made to vibrate two 
hundred degrees ; for, instead of having so much force as would 
compel the balance to vibrate over double the space with a double 



ISOCHRONISM OF BALANCE-SPRINGS. 17 

mean velocity, which would of course occupy the same time, it 
will possess an excess of tension, which will increase the velocity 
of each of the semi-vibrations, and necessarily abridge the time 
of performing them, and thus cause an accumulation of small 
instants, which will be the gain per diem. 

The remedy, in such a spring, is to increase the length of the 
part in action : but as this is not always convenient, or possible, 
the isochronal adjustment is resorted to, in which a kind of arti- 
ficial length is given to the spring, by compressing the curve of 
the part bent inward at the upper end, so as to make the curve 
commence its inward direction at a point a little farther distant 
from the stud.* 

Before attempting to make any alteration in a spring, it is 
advisable to examine the state of the curves, more especially 
when the chronometer gains in the long arcs, as it will sometimes 
be found, that one or other of the curves is turned in too abruptly, 
which has the effect of causing a gain in the long arcs, in conse- 
quence of the spring abutting so directly against the curve, as to 
leave a part of its length in very imperfect action. 

The opinion of the early ^^Titers upon isochronism, was, that a 
certain determinate length procured an isochronal spring ; but the 
fact is, that in every length of wire there are several isochronal 
points, to either of which a balance may be adapted, according 
to the nature of the vibrations it is intended to perform. Sup- 
pose, for instance, that a cylindrical spring, having ten turns, be 
found isochronal, one or more of these turns may be taken away, 
and a point in the spring still be found, that will give the 
required ratio of increasing tension, and produce isochronous 
vibrations. 

The spiral, or flat spring, is less simple in its form than the 
cylindrical spring ; and although whatever be the form, the prin- 
ciples upon which its isochronism depends are not altered, yet 
there are circumstances which affect its isochronal perfection in 
so material a degree, that they require to be particularly noted ; 
and the more especially so, since the spirals are more commonly 

* The old methods of tapering the spring, or thinning the upper or under turn, 
to malie the increase of tension observe the proper ratio, are never now resorted to 
by experienced makers. 



18 



ISOCHRONISM OF BALANCE-SPRINGS. 



employed than the cylindrical springs, while the construction of 
the former involves several points of great nicety in the manipu- 
lation. 

The proper length and strength of wire being well selected, 
the manner in which it is turned up into a spiral is important, for 
in this operation its natural isochronism may be either partially 
or wholly destroyed. This will especially happen if there be any 
small points, or elbows in it ; or if the spring be so made, that 
during the vibration, any part thereof be either inactive, or have 
an imperfect action. Indeed, the absolute necessity for the 
spring to continue in free and unrestrained action throughout its 
entire length, and diu'ing the whole period of the vibration, can- 
not be too strongly vu'ged ; because, an opinion generally prevails, 
that the outer turns do not come into action until near the end 
of the semi-vibrations. 

With the cylinch'ical spring, there is no great difficulty in pro- 
ducing the same extent of vibration on either side of the point 
of quiescence ; with a flat spring, however, this is not attained 
with an equal degi-ee of facility, nor without close attention to its 
form, as well as to the mode of pinning it in, so that it be not in 
the slightest degree strained from its natural shape, or position, 
when out of the watch. 

A spiral, to be turned up correctly, should lie in several close 
turns towards the centre (Fig. 2), springing off from where it is 
pinned into the collet, by a gentle curve, and thence gradually 
and constantly expanding, in such a manner that each part of the 
spiral would cross, but nowhere coincide with a small ciixular arc 
drawn from the centre of the collet and concentric therewith. 
This is indispensible to isochronism. 

Fie. 2. 




ISOCHRONISM OF BALANCE-SPRINGS 19 

If, on the contrary, a spiral springs off from the collet, first by 
a large bold sweep, and then lies in a few close and large turns, it 
Avill be very defective in its action, and quite devoid of the 
isochronal property. In such a spring, the middle of the vibra- 
tion will not coincide with the point of quiescence ; for the spring 
will readily yield to the momentum of the balance, during the 
winding up, and contraction of its coils, and the whole length of 
the spring will be brought into action, though imperfectly : but, 
during the expansion of the coils upon the return of the balance, 
the action of the inner turns will not be exerted against curves 
which lie across concentric chcles, as in the diagram, but against 
such as lie in concentric circles, or very nearly so, and will there- 
fore abut so point-blank against them, as to cause no displacement 
whatever in a portion of the outer turn ; thus giving the effect of 
a short strong spring, which arrests the balance too soon in this 
part of the vibration. Such irregularities are obviously quite in- 
compatible with the requisites for producing isochronous vibrations. 

The isochronal trial of a flat spring in a watch, is more simple 
than that described for the chi-onometer, since the balance for the 
watch is thrown into the long, or short arcs of vibration, by the 
mere change of position, which varies the amount of friction, and 
consequently the extent of the arc. In the horizontal position, 
with the face uppermost, the friction is least, and the vibrations 
are of the fullest extent ; in the vertical, or that in which the 
watch is worn, the friction is greatest, and the extent of the 
vibration is necessarily curtailed. The trial is made, with the aid 
of a good clock, by comparing the rate of performance during a 
certain number of hours, in a horizontal position, Math the mean 
result of an equal number of hours' performance in any two oppo- 
site vertical positions ; for instance, first with the xii. and then 
with the VI. upwards, and then in the like manner with the ix. 
and III. : the mean result of two opposite vertical positions being 
required, in order to neutralize the eff'ect of any slight irregulari- 
ties that may exist in the poise of the balance. The indications 
and the application of the isochronal adjustment, are the same as 
those already described for cylindrical springs, but under greater 
restrictions. For, as the balances for watches are, for the most 
part, simple and unprovided with any means by which their inertia 
3 



20 ISOCHRONISM OF BALANCE-SPRINGS. 

may be varied, as is done in the compensation balance, so as to 
suit the elastic force of any particular spring, and the number of 
vibrations required to be performed in a given time, the spiral of 
a watch must not only be isochronal, but be of the precise degree 
of elastic force demanded by the particular balance to be employ- 
ed. ■^•' The selection of a spring, within the limits of the isochro- 
nal adjustment, must therefore be made by trial in the watch. 

The great advantage of an isochronal spring, in its innate 
power of resisting the influences which cause a change of rate, 
such as the change of position, increased friction as the works 
become dirty, or the viscidity of the oil in low temperature. f 
Indeed, it is surprising to see chronometers return from sea with 
scarcely a change of rate, although some of them had been going 
for three or fom* years, and even for longer periods of time, and 
the vibrations had fallen off to a very small arc, in consequence 
of the oil becoming so viscid, that, in some instances, a slight 
degree of force has been required to draw the pivot out of the 
fourth wheel-hole. But what was still more remarkable, some of 
these chronometers, after being cleaned, had been observed to 
take up their original rate, even perhajjs with a threefold amount 
of vibration. 

The mode by which an isochronal spring arrives at such per- 
fection, may be thus explained : The spring's elastic force is 
presupposed to be both perfect and permanent, under similar 
temperatures ; for, as has been previously stated, the elastic force 
diminishes, as the temperature to which it is exposed is increased. 

The elastic force of the spring is counterbalanced by the resist- 
ance it meets with, in the work it has to jjerform ; this is of two 
kinds, the inertia of the balance, and the friction of the rubbing 
parts, from a certain amount of which no machinery is exempt. 

If the spring is assumed to possess a force equal to 100, and 
that 10 of these parts are requisite to overcome the friction, when 

* A compensation balance ia sometimes applied to a lever watch, with the design to 
render it a more perfect instrument; but that cannot be attained, unless duo regard be 
had to the isochronism of the spring, as will be understood from what has been already 
stated. 

t Watches of excellent workmanship occasionally come into the Author's hands, 
wilh considerable errors in the long or the short arcs of vibration ; yet they require 
nothing more than the isochronal adjustment to render them, what their makers in- 
tended, good watches. 



ISOCHRONISM OF BALANCE-SPRINGS. 21 

at a minimum, there will be 90 parts left, for action upon the 
balance. But the friction will vary according to circumstances, 
although the spring and balance may remain unaltered. If, there- 
fore, the spring has power to carry the balance through a circle 
of vibration, when the friction is at a minimum, it will have power 
to perform the same amount of work when the friction is at a 
maximum, but the 100 parts of force will be diffei-ently appor- 
tioned in the execution of the task. 

Let it be assumed, for instance, that the friction is trebled ; 
there will then be 30 parts employed in overcoming the friction, 
and consequently 70 parts only for action upon the balance, which 
will necessarily have a less extent of vibration. Now, since the 
isochronal ratio of the spring's tension remains unaltered, the be- 
ginning and end of each semi-vibration will still coincide with the 
first and last of the minute instants of time, composing the 
isochronous vibration, which is the condition that is required for 
correct performance. 

So likewise it is with increased friction in watch-work, the 
elastic forces of the balance spring being constantly proportioned 
to the angle of inflection, whatever the amount of friction, the 
law of isocln-onism remains unchanged, and friction is only an 
adventitious chcumstance, which aff'ects the extent of the arc of 
vibration, but not the time in which it will be described. 



Mr. Faket considered the Paper to be of a class highly de- 
serving of the attention of the Institution, for although the subject 
did not come within the ordinary course of engineering studies, 
nevertheless it involved principles of mechanical action, with 
which all engineers ought to be well acquainted. The application 
of those principles in the construction and operation of marine 
chronometers, had been attended with a degree of precision in 
■performance, greatly exceeding all that could be required in ma- 
chines for the performance of forcible operations ; those being the 
machines with which engineers were most conversant, and which 
they had brought to a high degree of perfection, as regarded 
power, strength, and rapidity of operation, in overcoming resist- 
ances, with capability of continuing their operation without 
ceasing, as long as might be requhed. 



22 ISOCHRONISM OF BALANCE-SPRINGS. 

It was a useful exercise of mind, to turn the thoughts occasion- 
ally from their habitual course, to the consideration of subjects, 
wherein similar principles had been applied with success, for the 
attainment of very different objects ; and it was one advantage of 
the Institution, that it brought together persons who had acquired 
skill in many different pursuits, having but little in common, ex- 
cept that they were all applications of mechanical science, and 
were dependent on the correctness of such applications for their 
success and theii- progress towards perfection. 

Chronometers required far more precision in then- motions, than 
any other machines, and the perfection to which they had been 
brought, rendered the principles of their construction an interest- 
ing study, and a correct knowledge of the very minute circum- 
stances on which that perfection was dependent, could not fail to 
be useful. Mr. Frodsham's paper explained, that isochronism of 
the vibration of the balance was a most essential qualification for 
correct performance in a chronometer, and that with some partic- 
ular length and other conditions of its balance-spring (only to be 
found by ti'ial in each case), the requisite equality of the times of 
long, or short arcs of vibration of the balance, might be attained. 

It was generally admitted as true in theory, that a spring which 
exerted an vmiformly increasing force, in being bended during the 
motion of tlie balance through equal arcs, and an miiformly 
decreasing force in unbending itself, and returning the balance 
through the same arcs, would produce the required regulation. 
As an illustration of this, the chronometer might be supposed 
to be placed M'ith its balance in a vertical plane, and a very 
fine hair to be fastened to its rim, and after passing around 
its circumference, descending vertically therefrom in the direc- 
tion of a tangent, so that any weight which might be appended 
to the hair wovdd move the balance some way round from its 
quiescent position, and in so doing would bend its spring, until 
the increasing force thereof becoming equal to react against 
the -weight, the balance M^ould then stop at some angle fi-om its 
quiescent position. If the spring were coiTCCtly formed, the ex- 
tent of svxch angle should be proportionate to the weight (id lensio 
sic vis, as Dr. Hooke expressed it). So that supposing the weight 
to be one grain, and that it had moved the balance through an 
angle of 20 degrees, then the addition of another grain should 



ISOCHRONISM OF BALANCE-SPRINGS. 23 

move the balance another 20 degrees; and so on, each increment 
of one grain of force should produce an increment of 20 degrees 
of angular motion of the balance ; and therefore 10 grains of force 
Avould move the balance through ten such increments of 20 
degrees each, or an angular motion of 200 degrees from the 
quiescent position, that being about an average extent of the 
semi-vibration of the balance in chronometers. The same mode 
of trial should be supposed to be repeated, with the hair passed 
in a contrary direction over the circumference of the balance, and 
ought then to show the same results as before ; thereby proving, 
the force of the spring to be the same, whether it was being 
wound up in the direction of its coils, or being unwound in a 
contrary direction. Such a condition of spring having been 
attained, in any watch or chronometer, the vibrations of its bal- 
ance would (according to theory) be performed in an equal space 
of time, whether they were of a longer, or shorter extent. There- 
fore a chronometer newly cleaned and oiled, being adjusted to 
keep time correctly, with its balance vibrating full one turn and a 
quarter (or 225 degrees for the semi-vibration) ought, not to alter 
its rates of going as the vibrations became diminished, in the 
course of constant use without oiling, even though they should be 
reduced to less than one turn (or 175 degrees for the semi-vibra- 
tion). 

The attainment of such an amount of isochronism was also of 
importance for pocket watches, although they did not require the 
precision of chi'onometers ; because, in addition to continuing to 
go well as the works became dirty, the fusee and chain for the 
main-spring might he dispensed with in the construction of the watch, 
and the diminution of the vibrations, attendant on the diminishing 
force of the main-spring, during each twenty-four hours, should 
make no difference in the going of the watch. 

The progressive motion of the balance, during its semi-vibra- 
tion, in returning from one extremity of the vibration to the 
quiescent, or midway position, was an accelerated motion ; and 
in proceeding onwards through the opposite semi-vibration beyond 
the midway position to the other extremity of the vibration, it 
was a retarded motion ; but the acceleration would not take place 
by the same law as that of falling bodies, because gravitation 
(practically speaking) was an uniform and constant force, which 



24 



ISOCHRONISM OF BALANCE-SPRINGS. 



gave to falling bodies equal increments of velocity in equal times, 
which was termed uniformly accelerated motion ; whereas the 
force of the balance spring was uniformly decreasing, and could 
not therefore produce so rapid an increment of velocity as uni- 
formly accelerated motion. 

Although the theory that isochronal vibrations must result from 
a uniformly increasing and decreasing force of the spring, might 
be admitted, yet, in practice, it was necessary that the spring 
should be adjusted, so that the law of its force would be accom- 
modated to suit all other circumstances, which could interfere 
Avith the freedom of the vibration of the balance, the resistance 
of the air being one of those circumstances : and in compensa- 
tion balances which had screws projecting from their circumfer- 
ences, with large heads to serve as weights, the resistance to their 
motion througli the an- must be considerable : but that resistance 
increasing as the squares of the velocity, it could not occur with 
the theoretically assumed uniformity of increasing and decreasing 
force of the balance spring, to allow of the isocln-onal vibrations, 
which it was admitted in theory, would result from such vmi- 
formity. 

The usual practice was to adjust to chronometer to go well, 
when its balance vibrated to a full extent of one revolution and a 
quarter, and then by letting down the main-spring by its rachet 
the impelling force was so diminished that the balance would not 
vibrate more than tkree-fourths of a revolution ; it was then 
expected that the chronometer would go as well as it did at first, 
and if it did not, the spring of the balance was altered in its 
length, or in the curvatm-e of its ends, in the manner described 
in the paper, until by repeated trials and alterations the chronom- 
eter was made to go at the same rate, whether the balance 
vibrated the full extent, or the diminished extent. 

This mode of trial was assumed to be an anticipation of what 
would afterwards take place, when the chronometer was in use on 
a long sea voyage, where it was found, that the extent of the 
vibrations of the balance diminished in proportion to the time the 
chronometer continued in use. But it might be doubted, on 
close examination of all the cu'cumstances, whether those of the 
previous trials, and those of the future use, were on a parity. 



ISOCHRONISM OF BALANCE-SPRINGS. 25 

Because when the maker reduced the extent of the vibrations for 
his trials, he did so by diminishing the impelling force that was 
imparted to the balance, without altering the resistance to the 
motion of the balance. Whereas in long service, the resistance 
to the motion of the balance became increased by the gradual 
thickening of the oil about its pivot, and that increased resistance 
was the chief cause of the diminishing extent of the vibrations. 

It was true that the oil in the wheelwork of the train would 
also thicken, and diminish the force with which the escapement 
wheel would impel the balance, so that both causes would be in 
operation, to diminish the extent of the vibrations, viz., greater 
resistance to motion, and less force of impulsion to overcome 
resistance, and produce motion ; but the influence of the thick- 
ening of the oil would would be much greater, to increase the 
resistance to the motion of the balance, than to diminish the 
impelling force of the train of wheelwork, because the friction of 
the pivots of the balance was caused by a very considerable 
motion between the rubbing surfaces, although attended with a 
very slight force of pressure: hence the viscidity of the oil, 
which was applied to those rubbing surfaces, would greatly aug- 
ment the resistance to motion ; whereas the friction of the wheel- 
work of the train was caused by a very small motion between the 
rubbing surfaces, attended with a considerable force of pressure, 
and an equal viscidity of the oil, would not diminish the impell- 
ing force of the wheelwork, in the same proportion as the resis- 
tance to the motion of the balance was increased. The trials 
which were made for adjusting the isochronism of the balance - 
spring of a chronometer, would be more assimilated to the cir- 
cumstances of its future xise, if a temporary friction could be 
applied, to retard the motion of the balance, as, for instance, by 
applying thick gummy oil to its pivots, or by means of a slender 
spring pressing latterally against the axis of the balance, and 
thereby reducing the extent of its vibrations, without depending 
entirely upon reduction of the impelling force of the main-spring 
and wheelwork for reducing that extent ; but both means might 
be resorted to, either separately, or in combination, for making 
the trials. 

It was stated in the Paper, that the friction of the balance did 
4 



26 ISOCHRONISM OF BALANCE-SPRINGS. 

not alter the isochronism of the spring, Avhen it had been fully 
attained. That might be the case, whilst the chronometer 
remained in the hands of the maker, after he had adjusted the 
spring by trial and error, as was described in the Paper ; because 
that adjustment, from the mode of making it, would compensate 
for the effect of all the diflerence of friction that must exist, when 
the motion of the balance was greater, or less, in the case when 
the diminution of the motion resulted solely from a reduction of 
the impelling force. But after the chronometer had been long in 
use at sea, and such diminution would chiefly be the result of 
increased resistance to the motion of the balance, from the thick- 
ening of the oil, there might be some doubt, whether the usual 
mode of making the adjustments would suit the future circum- 
stances as accurately as was desirable. 

The conditions under which a spring would produce isochronal 
vibrations of the balance were not well understood, although the 
result was arrived at by trial and error, as described in the Paper. 
An equality of flexure of every part of the spring throughout its 
whole length, if not a necessary condition, A^as no doubt very 
important, and would result most naturally from a tapered form 
of the spring, which was formerly given by working the steel for 
the spring thinner towards each of its ends. And although Mr. 
Earnshaw attributed much of the perfection of his chronometers 
to such tapering, it had not been continued, which might be 
because the tapering was very liable to be irregularly performed, 
and in such cases the superior accuracy of springs of uniform 
thickness would entitle them to a preference. 

Mr. Frodsham said there was no subject connected with 
horology, which had been the cause of so much loss of time, and 
waste of money and talent, as the prevailing want of knowledge 
of the laws of isochronism. Perfection of mechanism alone was 
incapable of insuring a correct time-keeper, or the detached 
escapements of Arnold and Earnshaw, which were models con- 
taining every requisite, would long ago have offered to the careful 
imitator, the means of producing perfect chronometers. But no 
mechanism, however correct in itself, unless under the influence 
of an isochronal spring, would produce a good result ; and per- 
sons who were unskilful in obtaining isochronism, had resorted to 



ISOCHRONISM OF BALANCE-SPRINGS. 27 

a variety of mechanical schemes to obtain arcs of uniform extent, 
and, as they hoped, of uniform duration. The remontoir escap3- 
ment was one in which, with much loss of power, a sort of 
secondary mainspring was wound up at short intervals, by the 
power of the primary spring. By this transfer of agency, all the 
errors of the mainspring and train were expected to be avoided, 
and it was argued, that as the impulse given to the balance would 
be constant and uniform, the arcs of vibration must, of necessity, 
be of iniiform extent, and therefore of uniform duration ; but it 
might easily be proved that one was not a necessary consequence 
of the other. There was, nevertheless, something so plausible in 
the remontoir escapement, that several talented men had labored 
to improve its construction, in a manner that did their persever- 
ence the highest credit ; for instance, Mudge, who effected the 
winding up of the remontoii' spring, at every half vibration, and 
it was to be regretted that so much zeal, talent and industry, 
should have been spent in fruitless research. 

The chronometer constructed Avith the Arnold escapement, con- 
tained adjustments for all the contingencies which it is liable to 
meet with at sea, and his late instruments were adjusted upon 
the simple isochi'onal principle, to maintain a uniform rate, in 
every variety of position ; notwithstanding they might occasion- 
ally fall off in their vibrations, to the extent of ninety degrees. 
How then could Mr. Farcy's setting up the main-spring be 
accomplished, or what necessity could there be for it, when the 
isochronal adjustment provided most perfectly for every change ? 
It would he dijficult indeed for an experienced hand to hazard any 
attempt at the adjustment of a chronometer at sea, when it ivas con- 
sidered that a variation of one degree of Fahrenheit' s thermometer 
produced in an uncompensated watch, an error of six seconds per 
diem. 

Mr. Farcy's remarks were exceedingly valuable, as to whether 
the mode adopted in reducing the arcs of vibration during trials, 
was really upon a parity with what took place by the thickening 
of the oils, after being at sea for a long period ; but Mr. Frod- 
sham's experience led him to believe that the adjustment had 
been found sufficient. 

The plan of tapering the springs was abandoned, because it 



28 ISOCHRONISM OF BALANCE-SPRINGS. 

was impossible to produce isochronism, under all circumstances, 
with a spring so made. In a chronometer it might be used, with- 
out detriment, nevertheless it distorted the action of the helical 
spring, Avhilst to the flat, or spiral spring, it gave a pleasing 
appearance during action ; but when ajiplied to the best lever 
watches it invariably produced a losing effect in the short arcs of 
vibration, amounting sometimes to one hundred and eighty sec- 
onds per diem. The plan was moreover unnecessary, because a 
more beautiful action of the spring might be obtained by bringing 
the outer turn of the spiral round, and pinning it in over the 
centre of the collet, or by a flexible stud ; as no error, however, 
arose from the thi-ow of the spring, it was best pinned into a 
fixed stud. 

Among the causes which aff'ected the extent of the arcs of 
vibration, were the dirty state of the watch, the thickening of the 
oil, and the fallings off" in the power of the mainspring, Avhich 
latter was sometimes as much as half its original force : yet if the 
balance-spring was isochronal, deviation from the original rate 
was generally found to be inconsiderable, even in such an extreme 
case. 

The lever watch Avas the most useful description of pocket 
watch for general wear ; yet as ordinarily constructed, it was well 
known to be very uncertain in its performance, and commonly 
lost in the short arcs of vibration (or vertical positions), to the 
extent of from one to four minutes per day. The cm-tailment of 
the vibrations and the thi-owing the balance out of poise, so as to 
accommodate the hanging and lying positions to each other, were 
usually resorted to ; whereas by the application of an isochronal 
spring, and a well-poised compensation balance of extended 
vibrations, excellent results might be obtained for pocket watches ; 
this should be generally known, because so much perfection had 
not hitherto been thought within the capabilities of the lever 
escapement. Few persons, except those Avho were in the con- 
stant habit of seeing the timing of chronometers under adjust- 
ment, could be aware of the difficulties attending the task ; and 
the wonder was, as an Astronomer Royal had remarked, that they 
" went so long and so well as they did." 

Another difficulty existed in the process of hardening and tem- 



ISOCHRONISM OF BALANCE-SPRINGS. 29 

pering the balance-spring, which was one of the most delicate 
operations connected with clironometry, and in which a few 
degrees difference in the temperature of the water into which it 
was plunged caused a difference in the state of hardness ; nor did 
the subsequent tempering obviate the effects to be observed, in 
the amount of gaining upon the rate of the instrument, which 
resulted from such hardness. 



We also append the following Table of Trains, by Charles 
Frodsham, showing that, with the same main-spring and the same 
balance-spring, nearly 70 different characters of balance may be 
used, by changing the velocity per minute, as per Table, which 
has a range of from 240 to 400 vibrations per minute. 

All have the same momemtum, and the same mean vibration 
between the hanging and lying position ; the arc of vibration 
agreeing more and more closely in the horizontal and vertical 
positions as the velocity per minute increases ; for, as a general 
rule, you cannot vary the diameter and keep a constant weight. 
It must also be well understood, that when we speak of momem- 
tum, it always has reference to the work done by a given main- 
spring and balance-spring, driving a balance of a certain diameter 
and weight a definite number of times in 60 seconds of mean 
time, through an arc of vibration M'hich is a correct mean between 
the horizontal and vertical positions, during an equal number of 
hours in each position. Thus, if a given main-spring and balance- 
spring carry a balance whose diameter is 0.65 inches and weight 
9 gi'S. through a certain arc of vibration 270 times per minute of 
mean time, the same main-spring and balance-spring will carry a 
balance of 0.65 in. diameter and 4.10 grs. weight 400 hundred 
vibrations per minute. But, if we prefer weight to diameter, 
then we can only use 0.44 inches diameter with a weight of 9 grs. 

From the above it Avill be seen a variety of changes can be 
made, the law being that the weight be increased or diminished 
in the inverse ratio of the square of the diameter, the square of 



30 ISOCHRONISM OF BALANCE-SPRINGS. 

the number of vibrations per minute, or the square of the number 
of degrees in the arc of vibration. 

The most useful trains are : — for lever escapements, a velocity 
of 270, 280 and 288 per minute; for marine chronometers, 240, 
the well-known half second beat ; for duplex, 300 per minute ; 
for pocket chronometers, 300 to 320. 

The question may probably arise, is there any time-keeping 
quality in any particular train ? None, or very little, except in 
extreme cases. Thus, for a stationary instrument (like the ship 
chronometer) the slow half second train is preferred ; but this 
would be quite impracticable in a watch that is to be carried in 
the pocket, as the balance-spring would not probably control its 
motion ; besides which, the arc of vibration in the horizontal and 
vertical positions, would disagree to too great an extent. 

On the other hand, in very high velocities, the properties of 
the balance are lost by the diminished weight or diameter, or both, 
and the balance has a constant tendency to be overmastered by 
the spring and be brought to rest. 



ISOCHE.ONISM OF BALANCE-SPRINGS. 



31 



GENERAL TABLE OF TRAINS. 



From 240 to 400 vibrations per minute. 

The measures are in lOOths of an in. and the weight in grs. 

The arc of vibration is constant, one turn and a half. 









- 


a ° 




o 












Vibrations. 




JS 




m 


2 




P 














^ 


K O 


C 


•§=^ 




^A 


Both va- 








Vibrations 
per second. 


Id 


^ii 


c 
o 




o 


II 


riable. 


Balances. 








Per 


Per 


Per 




&"" 


ot'^ 


" 


sr..S 


"ti 


. s 








hour. 


min. 


sec'nd. 




W 


pi 





'^ 


S 


5 


Wt. 


Dia. 




14,400 


240 


4 


4 V. = 1" 


12 


ItolO 


65 


11:34 


9 


73 


10 't 


68^ 


^ 


" 


" 


" 


" 


15 


1" 8 


65 


11:34 


9 


73 


lOi 


68i 


D. 681 


" 


• ' 


" 


" 


16 


2 " 15 


65 


11:34 


9 


73 


lOJ 


68i 


.W. 10 J 


" 


" 


" 


-' 


18 


3" 20 


65 


11:34 


9 


73 


lOV 

io| 


68§ 


grs. 


" 


" 


" 


" 


20 


1" 6 


65 


11:34 


9 


73 


681 




15,000 


250 


41-6 


25 V. = 6" 


15 


3 " 25 


65 


10:48 


9 


70J 


n 


671 




15,120 


252 


4 1-5 


21 V. = £" 


14 


1" 9 


65 


10::B2 


9 


692 


05 


673 




" 


" 


" 


" 


18 


1 " 7 


65 


10:32 


9 


695 


9| 


673 




15,360 


256 


4 4-15 


64 V. =15" 


16 


1" 8 


65 


10:00 


9 


68| 


81 


m 




15,600 


260 


4 1-3 


13 V. = S" 


13 


1 " 10 


65 


9:70 


9 


671 


9i 


663 




" 


" 


" 


" 


20 


2" 13 


65 


9:70 


9 


67| 


04 


663 

66| 




15,660 


261 


4 7-20 


87 V. ='2C" 


18 


4 " 29 


65 


9.62 


9 


67J 


9| 
9* 




15,840 


264 


4 2-5 


22 V. = 5" 


18 


3" 22 


65 


9:40 


9 


66J 


65s 




*16,200 


270 


41-2 


9 «;. = 2" 


lit 


1" 9) 
2 "15 


65 


9:00 


9 


65 


9 


65 


(B. 65 
\ W.ggrs. 


16,320 


272 


4 8-15 


68 V. =15" 


16 


2 " 17 


65 


8:87 


9 


64| 


9 


64i 




" 


" 


" 


" 


17 


1" 8 


65 


8:87 


9 


U\ 


9 


64i 




16,560 


276 


4 3-5 


23 V. = 5" 


18 


3 " 23 


6) 


8:60 


9 


63f 


H 


641 




10,800 


280 


4 2-3 


14 V. = o'l 


15 


3" 2S 


65 


8:38 


9 


62| 


8| 


63i 




" 


" 


" 


" 


16 


4" 35 


65 


8:36 


9 


62* 


4 


63 i 




" 


" 


" 


" 


20 


1" 7 


65 


8:36 


9 


62| 


8| 


631 




17,283 


288 


4 4-5 


24 V. = 5" 


16 


1" 9 


65 


7:90 


9 


61 


4. 


62| 




" 


" 


" 


" 


18 


1" 8 


65 


7:90 


9 


61 


8i 


62? 




17,340 


283 


4 49-60 


289 V. =3C" 


17 


2 "17 


65 


7:85 


9 


60? 


% 


62I 




17,400 


230 


4 5-6 


29 V. = 6" 


15 


3 "29 


65 


7:80 


9 


601 


84 
8| 


621 




" 


" 


" 


" 


20 


4" 29 


65 


7:80 


9 


601 


62^ 




18,000 


300 


5 


5v. = 1" 


12 


2 " 25 


65 


7:28 


9 


58i 


8| 


6lj 


-1 


" 


" 


" 


'' 


15 


1"10 


65 


7:28 


9 


581 


8i 


61 J 


D.61i 


" 


" 


" 


" 


16 


8 " 75 


65 


7:28 


9 


581 


H 


61i 


Uv.81 


" 


" 


" 


" 


18 


3" 25 


65 


7:28 


9 


58| 


8| 


61J 


grs. 


" 


" 


" 


" 


20 


2 " 15 


65 


7:28 


9 


58.1 


6U 


J 


18,440 


304 


5 1-15 


76 V. =15" 


16 


2" 19 


65 


7:10 


9 


57J 


84 


60? 




18,3)0 


303 


5 1-10 


51 V. =lt" 


18 


2 "17 


65 


7:00 


9 


57| 


8 


6O3 




18,600 


310 


5 1-6 


31 V. =16" 


15 


3 "31 


65 


6:83 


9 


56| 


8 


60' 




18,720 


312 


5 1-5 


26 V. = 5" 


18 


3 " 23 


65 


6:73 


9 


56 '( 


8 


59? 




19,200 


320 


5 1-3 


16 V. = o" 


15 


3 " 32 


65 


6:40 


9 


545 


8 


58', 




" 


" 


" 


" 


16 


1" 10 


65 


6:40 


9 


54J 


8 


58| 
581 




" 


" 


" 


" 


20 


1 " 8 


65 


6:40 


9 


bil 


8 




19,440 


324 


5 2-5 


27 V. = 5" 


18 


1 " 9 


65 


6:23 


9 


54i 


8 


571 




19,680 


328 


5 7-15 


82r;. =1" 


16 


4 "41 


65 


6:10 


9 


531 


■^4 


56? 




20,400 


340 


5 2-3 


17 V. = VI 


17 


1 " 10 


65 


5:67 


9 


51j 


71 


561 




• ' 


" 


" 


'' 


20 


4 " .34 


65 


567 


9 


513 


n 


561 


ID. 55 
I \V.7igs. 


21,000 


350 


5 5-6 


35 V. = C" 


20 
118 


4" 35 

1 " 10 ) 


65 


5:36 


9 


50J 


n 


55 


21,600 


S60 


6 


6 V. = 1" 


J20 


1 " 91 1 


65 


5:06 


9 


48:; 


l\ 


54J 




22,800 


38') 


6 1-3 


19 V. = 3" 


20 


2 " 19 


65 


4:55 


9 


461 


7 


52I 




24,000 


400 


6 2-3 


20 V. = 3" 


20 


1" 10 


65 


4:10 


9 


44' 


63 


50? 





• Model. 



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